Now, in the figure, M N = √(1−x2) and S N = 1+x; con hyp. log. cotan. P S M ; Therefore, L= or= hyp. log. tan. S M N, or or again, L= hyp. log. cotan. PC M, that is, in the projection, the length of a meridional arc measured from the equator, northwards or southwards (the latitude of its extremity being expressed by 1) is equal to the hyp. log. of the cotangent of half the complement of 7, the radius of the sphere being unity. This theorem was first demonstrated by Dr. Halley. (Phil. Trans., No. 219.) The numbers in the tables of meridional parts which are usually given in treatises of navigation may be obtained from the above formula; but in those tables the length of an equatorial minute is made equal to unity, and consequently the radius of the sphere is supposed to be 3437-75: therefore the value of L which is obtained immediately from the formula, must be multiplied by this number in order to have that which appears in the tables. For example, let it be required to find the number in the table of meridional parts corresponding to the 80th degree of latitude: Half the colatitude is 5°, whose log. cotan. =1.05805, and the logarithm of this number is Subtract the log. of modulus (0-43429) Log. of 8375 (=L) And 8375 is the number in the tables. 0.02451 -1.63778 0.38673 3.53627 3.92300 55. As an example of the manner in which Mercator's projection is applied, let the distance which a ship has sailed be 100 miles, on a course making an angle of 50° with the meridian of her point of departure; the latitude of this point being 60°. Let A be the point of departure, and draw the straight line PA to represent the meridian. Make the angle PAB equal to 50°; and, from any scale of equal parts representing geographical miles or equatorial minutes, take 100 for the length of AB, then draw B C perpendicular to PA. The line AC computed by plane trigonometry, or measured on the same scale, will express the difference of latitude (=64·28, or с P B 1° 4.28′), and in like manner CB(=76·6 or 1° 16.6') the Departure. Now the latitude of A being 60°, that of B or C is 61° 4.28', and from a table of meridional parts we have c for 61° 4.28', the number 4658: for 60°, the number 4527: the difference (=131) is the difference of latitude in the projection; therefore make A c, from the same scale, equal to 131, and draw cb parallel to CB. Then cb being computed, or measured on the scale, will be 156.15 or 2° 36.15', the difference between the longitudes of A and B. A CHAP. III. SPHERICAL TRIGONOMETRY. DEFINITIONS AND THEOREMS. 56. THE objects principally contemplated in propositions relating to the elementary parts of practical astronomy, are the distances of points from one another on the surface of an imaginary sphere, to which the points are referred by a spectator at its centre, and the angles contained between the planes of circles cutting the sphere and passing through the points, it being understood that the plane of a circle passing through every two points is intersected in two different lines by the planes of the circles passing through those points and a third. Thus the circular arcs connecting three points are conceived to form the sides of a triangle on the surface of the sphere; and hence the branch of science which comprehends the rules for computing the unknown sides and angles is designated spherical trigonometry. 57. In general, each side of a triangle is expressed by the number of degrees, minutes, &c., in the angle which it subtends at the centre of the circle of which it is a part, and each angle of the triangle by the degrees, &c. in the angle at which the two circles containing it are inclined to one another; but it is frequently found convenient to express both sides and angles by the lengths of the corresponding arcs of a circle whose radius is unity, the trigonometrical functions (sines, tangents, &c.) of the sides and angles being also expressed as usual in terms of a radius equal to unity. The latter method is absolutely necessary when any such function is developed in a series of terms containing the side or angle of which it is a function. It is obvious that, in order to render the measures of the sides of spherical triangles comparable with one another when expressed in terms of their radii, those radii must be equal to one another; and therefore such triangles are, in general, conceived to be formed by great circles of the sphere on whose surface they are supposed to exist. In the processes of practical astronomy, it is however often necessary to determine the lengths of the arcs of small circles as indirect means of finding the values of some parts of spherical triangles; but, before a final result is obtained, these must be converted into the corresponding arcs of great circles: occasionally also it is required to convert the arcs of great circles into the corresponding arcs of small circles, and the manner of effecting such conversions will be presently explained. 58. When the apparent places of celestial bodies are referred to what is frequently called the celestial sphere, whose radius may be conceived to be incalculably great when compared with the semidiameter of the earth, or with the distance of any planet from the sun, the arcs of great circles passing through such places may be supposed, at pleasure, to have their common centre at the eye of the spectator, or at the centre of the earth or of the sun; and the first supposition is generally adopted in computations relating to the positions of fixed stars: but, as the semidiameter of the earth is a sensible quantity, when compared with the distances of the sun, moon, and planets from its centre, and from a spectator on its surface, it is in general requisite, in determining the positions of the bodies of the solar system by the rules of trigonometry, to transfer those bodies in imagination to the surface of a sphere whose centre coincides with that of the earth. The planets, among which the earth may be included, are also conceived to be transferred to the surface of a sphere whose centre is that of the sun. 59. It is easy to perceive that there must be a certain resemblance between the propositions of spherical trigonometry and those which relate to plane triangles; and, in fact, any of the former, except one, may be rendered identical with such of the latter as correspond to them in respect of the terms given and required, by considering the rectilinear sides of the plane triangles as arcs of circles whose radii are infinitely great, or, which is the same, by considering them as infinitely small arcs of great circles of a sphere, whose radius is finite. For, in either case, on comparing the spherical triangles with the others, the sides of the former, if expressed as arcs in terms of the radius, may be substituted for their sines or tangents; also unity, or the radius, may be substituted for the cosines of the sides and the reciprocals of the sides for their cotangents. The exception alluded to is that case in which the three angles of a spherical triangle are given, to find any one of the sides; for in the corresponding proposition of plane trigonometry, the ratio only of the sides to one another can be determined; it may be observed, however, that the sides of a spherical triangle, when computed by means of the angles, are also indeterminate unless it be considered that they appertain to a sphere whose diameter is given. With this exception, the propositions of plane trigonometry might be considered as corollaries to those of spherical trigonometry for the case in which the spherical angles become those which would be made by the intersections of three planes at right angles to that which passes through the angular points of the triangle, and in which, consequently, they are together equal to two right angles only; and it is evident that the sum of the angles of a spherical triangle of given magnitude approaches nearer to equality to two right angles as the radius of the sphere increases. PROPOSITION I. 60. To express one of the angles of a spherical triangle in terms of the three sides. C N B Let ACB be a triangle on the surface of a sphere whose centre is o, and imagine its sides to lie in three planes passing through that centre intersecting one another in the lines OA, OB, OC. (The inclinations of the planes to one another, or the angles of the spherical triangle, are supposed to be less than right angles, and each of the sides to be less than a quadrant.) Imagine a plane to pass through oc perpendicularly to the plane AOB, cutting the surface of the sphere in CD: then the triangle ABC will be divided into two right-angled triangles P ADC, BDC. Next let MCN be a plane touching the sphere at C, and bounded by the planes COA, COB, BOA produced, and let it meet the latter in MN; also imagine the plane ODC to be produced till it cuts MCN in CP. The plane COP is by supposition at right angles to MON; and because CO is perpendicular to the tangent plane MCN, the plane COP is perpendicular to the same plane MCN; therefore (Geom. 19. Planes, and 1. Def. Pl.) MN is perpendicular to the plane COP and to the lines OP, CP: hence the plane triangles CPM, CPN, OPM, OPN are right angled at P. Now (Plane Trigon., art. 56.) |